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Generalizations of "standard" calculus
This could be an answer for your last question: if you define $\frac{\Delta_t}{t}$ as $\frac{f(x+t)-f(x)}{t}$ for $t>0$ and you solve $\frac{\Delta_t}{t}(f)=f$ you obtain the solution $$f(x)=\left((1+ …